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We study SO(4) BF theory plus a general quadratic potential, which describesa bi- metric theory of gravity. We identify the profile of the potentialleading to a Pauli-Fierz mass term for the massive graviton, thereby avoidingthe linearized ghost. We include the Immirzi parameter in our analysis, andfind that the mass of the second graviton depends on it. At thenon-perturbative level, we find a situation similar to genuine bi-gravities:even choosing the Pauli-Fierz mass term, the ghost mode propagates through theinteractions. We present some simple potentials leading to two and threedegrees of freedom, and discuss the difficulties of finding a ghost-freebi-gravity with seven degrees of freedom. Finally, we discuss alternativereality conditions for the case of SO(3,1) BF theory, relevant for Lorentziansignature, and give a new solution to the compatibility equation.

Pauli-Fierz mass term in modified Plebanski gravity